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The large sieve inequality

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Tools on Fourier transforms

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The best version of the large sieve inequality from and (obtained at the same time by A. Selberg) is as follows.
Theorem (1974)

Let $M$ and $N\ge 1$ be two real numbers. Let $X$ be a set of points of $[0,1)$ such that $ \displaystyle \min_{\substack{x,y\in X\\ x\neq y}} \min_{k\in\mathbb{Z}}|x-y+k|\ge \delta>0. $ Then, for any sequence of complex numbers $(a_n)_{M < n\le M+N}$, we have $$ \sum_{x\in X}\left| \sum_{M < n\le M+N} a_n \exp(2i\pi nx) \right|^2 \le \sum_{M < n\le M+N}|a_n|^2 (N-1+\delta^{-1}). $$

It is very often used with part of the Farey dissection.
Theorem (1974)

Let $M$ and $N\ge 1$ be two real numbers. Let $Q\ge1$ be a real parameter. For any sequence of complex numbers $(a_n)_{M < n\le M+N}$, we have $$ \sum_{q\in Q}\sum_{\substack{a\mod q,\\ (a,q)=1}}\left| \sum_{M < n\le M+N} a_n \exp(2i\pi na/q) \right|^2 \le \sum_{M < n\le M+N}|a_n|^2 (N-1+Q^2). $$

The summation over $a$ runs over all invertible (also called reduced) classes $a$ modulo $q$. The weighted large sieve inequality from Theorem 1 in reads as follows.
Theorem (1974)

Let $M$ and $N\ge 1$ be two real numbers. Let $X$ be a set of points of $[0,1)$. Define $ \displaystyle \delta(x)=\min_{\substack{y\in X\\ y\neq x}} \min_{k\in\mathbb{Z}}|x-y+k|. $ Then, for any sequence of complex numbers $(a_n)_{M < n\le M+N}$, we have $$ \sum_{x\in X}\left| \bigl(N+c\delta(x)^{-1}\bigr)\sum_{M < n\le M+N} a_n \exp(2i\pi nx) \right|^2 \le \sum_{M < n\le M+N}|a_n|^2 $$ for $c=3/2$.

It is expected that one can reduce the constant $c=3/2$ to 1. In this direction, we find in the next result.
Theorem (1984)

We may take $\displaystyle c= \sqrt{1+\tfrac23\sqrt{\tfrac65}} = 1.3154\cdots < 4/3$ in the previous theorem.

See for a discussion on this topic.

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